315-317] OTHER FORMS OF EQUILIBRIUM. 589 



were the crossing-point with another linear series. For this 

 reason it is called a form of bifurcation. A great part of 

 Poincare s investigation consists in ascertaining what members of 

 Maclaurin s and Jacobi s series are forms of bifurcation. 



Poincare also discusses very fully the question of stability, to 

 which we shall briefly revert in conclusion. 



317. The motion of a liquid mass under its own gravitation, 

 with a varying ellipsoidal surface, was first studied by Dirichlet*. 

 Adopting the Lagrangian method of Art. 13, he proposes as the 

 subject of investigation the whole class of motions in which the 

 displacements are linear functions of the velocities. This has 

 been carried further, on the same lines, by Dedekindf and 

 RiemannJ. More recently, it has been shewn by Greenhill and 

 others that the problem can be treated with some advantage by 

 the Eulerian method. 



We will take first the case where the ellipsoid does not change 

 the directions of its axes, and the internal motion is irrotational. 

 This is interesting as an example of finite oscillation of a liquid 

 mass about the spherical form. 



The expression for the velocity-potential has been given in 

 Art. 107 ; viz. we have 



with the condition of constant volume 



%M = 

 a b c 



The pressure is then given by 



* &quot; Untersuchungen iiber ein Problem der Hydrodynamik,&quot; Gott. Abh., t. viii. 

 (1860) ; Crelle, t. Iviii. The paper was posthumous, and was edited and amplified 

 by Dedekind. 



t Crelle, t. Iviii. 



J &quot;Beitrag zuden Untersuchungen iiberdieBewegung ernes fliissigen gleicharti- 

 gen Ellipsoides,&quot; Gott. Alh., t. ix. (1861); Math. Werke, p. 168. 



&quot;On the Rotation of a liquid Ellipsoid about its Mean Axis,&quot; Proc. Camb. 

 Phil. Soc. t t. iii. (1879); &quot;On the general Motion of a liquid Ellipsoid under the 

 Gravitation of its own parts,&quot; Proc. Camb. Phil. Soc., t. iv. (1880). 



